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GKP codes encode a qubit in displaced phase space combs of acontinuous-variable (CV) quantum system and are useful for correcting a varietyof high-weight photonic errors. Here we propose atomic ensemble analogues ofthe single-mode CV GKP code by using the quantum central limit theorem to pullback the phase space structure of a CV system to the compact phase space of aquantum spin system. We study the optimal recovery performance of these codesunder error channels described by stochastic relaxation and isotropic ballisticdephasing processes using the diversity combining approach for calculatingchannel fidelity. We find that the spin GKP codes outperform other spin systemcodes such as cat codes or binomial codes. Our spin GKP codes based on thetwo-axis countertwisting interaction and superpositions of SU(2) coherentstates are direct spin analogues of the finite-energy CV GKP codes, whereas ourcodes based on one-axis twisting do not yet have well-studied CV analogues. Animplementation of the spin GKP codes is proposed which uses the linearcombination of unitaries method, applicable to both the CV and spin GKPsettings. Finally, we discuss a fault-tolerant approximate gate set for quantumcomputing with spin GKP-encoded qubits, obtained by translating gates from theCV GKP setting using quantum central limit theorem.

Spin squeezed GKP codes for quantum error correction in atomic ensembles